Theory Of Integrable Systems Research Articles

Q-operators are ’t Hooft lines

We study ’t Hooft lines in four-dimensional holomorphic-topological Chern-Simons theory. We relate them to Q-operators in the theory of integrable systems. We give a physical interpretation of the fundamental TQ and QQ relations satisfied by Q-operators and conventional transfer matrices.

From quantum field theory to the quantum method the inverse problem On the 90th anniversary of the birth of academician L.D. Faddeev

The article is dedicated to the life and work of a theoretical physicist and mathematician, a full member of the USSR Academy of Sciences and RAS L.D. Faddeev (1934−2017), the formation of his scientific school. Faddeev’s works on scattering theory, quantum field theory, classical and quantum theory of integrable systems were included in the golden fund of world science and largely determined the face of modern mathematical physics. The author of the article is an employee of the Laboratory of Mathematical Problems of Physics created by L.D. Faddeev at the St. Petersburg Department of the V.A. Steklov Mathematical Institute of the Russian Academy of Sciences since its foundation.

Nonexistence of integrable nonlinear magnetic fields with invariants quadratic in momenta

Nonlinear, completely integrable Hamiltonian systems that serve as blueprints for novel particle accelerators at the intensity frontier are promising avenues for research, as Fermilab’s Integrable Optics Test Accelerator (IOTA) example clearly illustrates. Here, we show that only very limited generalizations are possible when no approximations in the underlying Hamiltonian or Maxwell equations are allowed, as was the case for IOTA. Specifically, no such systems exist with invariants quadratic in the momenta, precluding straightforward generalization of the Courant–Snyder theory of linear integrable systems in beam physics. We also conjecture that no such systems exist with invariants of higher degree in the momenta. This leaves solenoidal magnetic fields, including their nonlinear fringe fields, as the only completely integrable static magnetic fields, albeit with invariants that are linear in the momenta. The difficulties come from enforcing Maxwell equations; without constraints, we show that there are many solutions. In particular, we discover a previously unknown large family of integrable Hamiltonians.

Darboux transformation-based LPNN generating novel localized wave solutions

Darboux transformation method is one of the most essential and important methods for solving localized wave solutions of integrable systems. In this work, we introduce the core idea of Darboux transformation of integrable systems into the Lax pairs informed neural networks (LPNNs), which we proposed earlier. By fully utilizing the data-driven solutions, spectral parameter and spectral function obtained from LPNNs, we present the novel Darboux transformation-based LPNN (DT-LPNN). The notable feature of DT-LPNN lies in its ability to solve data-driven localized wave solutions and spectral problems with high precision, and it also can employ Darboux transformation formulas of integrable systems and non-trivial seed solutions to discover novel localized wave solutions that were previously unobserved and unreported. The numerical results indicate that, by utilizing the single-soliton solutions as the non-trivial seed solutions, we obtain novel localized wave solutions for the Kraenkel–Manna–Merle (KMM) system by employing DT-LPNN, in which solution u changes from original bright single-soliton on zero background wave to new dark single-soliton dynamic behavior on a variable non-zero background wave. Moreover, by treating the two-soliton solutions as the non-trivial seed solutions, DT-LPNN generates novel localized wave solutions for the KMM system that exhibit completely different dynamic behaviors from prior two-soliton solutions. DT-LPNN combines the Darboux transformation theory of integrable systems with deep neural networks, offering a new approach for generating novel localized wave solutions using non-trivial seed solutions.

Development and Analysis of Novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Two-Component Nonlinear System with the On-Site and Spatially Distributed Inertial Mass Parameters

The main principles of developing the evolutionary nonlinear integrable systems on quasi-onedimensional lattices are formulated in clear mathematical and physical terms discarding the whimsical mathematical formulations and computer-addicted presentations. These basic principles are substantiated by the actual development of novel semi-discrete integrable nonlinear system, whose auxiliary spectral and evolutionary operators are given by 4 × 4 square matrices. The procedure of reduction from the prototype nonlinear integrable system with twelve field functions to the physically meaningful nonlinear integrable system with four field functions is described in details prompted by our previous cumulative experience. The obtained ultimate semi-discrete nonlinear integrable system comprises the two subsystems of essentially distinct physical origins. Thus, the first subsystem is the subsystem of the Toda type. It is characterized by the on-site (spatially local) mass parameter and the positively defined elasticity coefficient. In contrast, the second subsystem is characterized by the spatially distributed mass parameters and the negatively defined elasticity coefficient responsible for the low-amplitude instability. We believe our scrupulous consideration of all main steps in developing the semidiscrete nonlinear integrable systems will be useful for the researchers unfamiliar with the numerous stumbling blocks inevitable in such an interesting and prospective scientific field as the theory of semi-discrete nonlinear integrable systems.

Nonisospectral water wave field: Fast and adaptive modal identification and prediction via reduced-order nonlinear solutions.

Real-world water wave fields exhibit significant nonlinear and nonisospectral characteristics, making it challenging to predict their evolution by relying solely on numerical simulation or exact solutions using integrable system theory. Hence, this paper introduces a fast and adaptive method of modal identification and prediction in nonisospectral water wave fields using the reduced-order nonlinear solution (RONS) scheme. Specifically, we discuss the coarse graining and mode extraction of wave field snapshots from the data-driven and physics-driven perspectives and utilize the RONS method for principle modal prediction of nonisospectral water wave fields. This is achieved by investigating the standard and nonisospectral Gardner system describing nonlinear water waves as a demonstration. Through detailed comparison and analysis, the fundamental solitary behaviors and dispersive effects in the Gardner system are discussed. Subsequently, a neighbor approximation is developed that combines the essences of symbolic precomputation and numerical computation in the RONS procedure, which exploits the locality of nonlinear interactions in water wave fields.

Integrability and Dynamic Behavior of a Piezoelectro-Magnetic Circular Rod

The present work strives to explore some qualitative analysis for the governing equation describing the dynamic response of a piezoelectro-magnetic circular rod. As a result of the integrability study of the governed equation, which furnishes valuable insights into its structure, solutions, and applications in various fields, we apply the well-known Ablowitz–Ramani–Segur (ARS) algorithm to prove the non-integrability of the governed equation in a Painlevé sense. The qualitative theory for planar integrable systems is applied to study the bifurcation of the solutions based on the values of rod material properties. Some new solutions for the governing equation are presented and they are categorized into solitary and double periodic functions. We display a 3D representation of the solutions in addition to investigating the influence of wave velocity on the obtained solution for the particular material of the rod.

Algebraic de Rham theorem and Baker-Akhiezer function

For the case of algebraic curves (compact Riemann surfaces), it is shown that de Rham cohomology group $H^1_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$ of the Riemann surface $X$ has a natural structure of a symplectic vector space. Every choice of a non-special effective divisor $D$ of degree $g$ on $X$ defines a symplectic basis of $H^1_{\mathrm{dR}}(X,\mathbb{C})$ consisting of holomorphic differentials and differentials of the second kind with poles on $D$. This result, which is the algebraic de Rham theorem, is used to describe the tangent space to Picard and Jacobian varieties of $X$ in terms of differentials of the second kind, and to define a natural vector fields on the Jacobian of the curve $X$ that move points of the divisor $D$. In terms of the Lax formalism on algebraic curves, these vector fields correspond to the Dubrovin equations in the theory of integrable systems, and the Baker-Akhierzer function is naturally obtained by the integration along the integral curves.

Preface: Application of Mathematics in Technical and Natural Sciences 15th International Conference - AMiTaNS’23

The Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences organized its 15th annual conference of the series AMiTaNS (Application of Mathematics in Technical and Natural Sciences). The event took place during June 21-26, 2023 in the Congress Centre of Black-Sea resort of Albena, Bulgaria. Its aim was to foster existing and new personal contacts, scientific discussion as well as to provide a forum for the exchange of ideas and experience between specialists in the fields of Mathematics, Physics, Biology, Ecology, Economics, Computer Science and Technology. The event became a world conference covering 15 time zones between Chile and Nebraska through Texas, Indiana, Alabama, Florida, the UK, Europe, Cyprus, Russia, and China. The conference was managed from the Congress Centre in Albena where 40 participants and delegates were present in person and the other 40 attended the event via Zoom video platform.Interest in this conference is growing year by year. This year we registered 70 talks and scientific reports, more than 80 participants – contributors and delegates from 16 countries around the world, representing the continents of Europe, North and South America, and Asia. About 18% of them attended the conference for the first time, with 50% of them being young scientists. The accepted talks were delivered and attended both on-location at the big hall and Corolla hall in Flamingo Grand hotel and on-line. Every day the morning sessions started by plenary lectures followed by contributed sessions on the following topics: Applied Analysis, Biomathematics, Continuum Mechanics, Mathematical Physics, Numerical Methods, Scientific Computing. As part of the program, leaders of their respective fields organized three special sessions: Statistics & Modeling, Mathematical Methods in Reliability Theory, Algebraic and Geometric Aspects of Integrable Systems. A number of very interesting presentations were delivered, which sparked stimulating scientific discussions both on- and off-line. The reader can find comprehensive and detailed information about the talks and reports including their records on the conference website at http://2023.eac4amitans.eu.All contributed manuscripts were reviewed by anonymous independent referees via Morressier platform, and the approved revised manuscripts were accepted for publication in the present volume. The continued success of AMiTaNS in defiance of the extraordinary circumstances and challenges faced in the last few years undoubtedly enhances its reputation as an established annual meeting with an eminent and continuously growing core of contributors. Yet, the hybrid format turned out very flexible both for the organizers and contributors. It makes the conference accessible to all those who wish to attend either in person or remotely. We would like to thank the plenary speakers, who accepted our invitation and contributed to the conference through their elucidating lectures. We would also like to thank the special sessions organizers, hosts and co-hosts for the high scientific quality of the talks they selected, and for ensuring the smooth running of the conference in hybrid format. We are deeply indebted to the scores of anonymous referees all over the world, who approached their duties very professionally and thoroughly, and helped us to maintain high scientific standards. We highly appreciate the involvement of the members of the Program Committee and Steering Committee, who supported AMiTaNS from its inception, and worked tirelessly throughout the intensive labors of advertising, arranging, organizing, and reviewing. Special thanks go to our colleagues from the University of Essex at Colchester, UK, who worked with dedication so that parallel Zoom conference sessions could be held completely on-line.We are very grateful to our publisher Institute of Physics, Bristol, UK for the helpful cooperation during the preparation of this volume.List of Program Committee, Organizer, Sponsor, Participants are available in this Pdf.

Impacts of Information technology and Risk Management on Cybersecurity Governance: Empirical Study on Malaysian Financial Institutions

"Cybersecurity threats have successfully targeted financial institutions worldwide due to the increased connectivity of seamless and borderless financial services. Considering that criminals have employed more sophisticated methods to exploit the financial industry, financial institutions must adopt an integrated framework to counter the attacks and protect financial infrastructure from exploitation. The study examined cybersecurity governance by extending the Integrated System Theory (IST), including information technology (IT) governance and risk management (RM) governance. Questionnaires were distributed to Malaysian financial institutions through corporate social media platforms, email, and Google Form in November 2021. The questionnaire used a five-point Likert scale and comprised Section A which focused on the respondents’ demographic profile, while Section B emphasised the research construction. A total of 128 respondents participated within four weeks. Data obtained from the questionnaire was analysed for descriptive statistics, correlation, and regression analyses. Resultantly, IT governance and RM governance significantly and positively impacted cybersecurity governance. The study provides better insights to the practitioners, academicians, or researchers to identify which factors to be considered and emphasise before developing an integrated cybersecurity governance framework."

A new look into the concept of separable system of Classical Mechanics

The theory of integrable systems of Classical Mechanics has been deeply affected, in the second half of the last century, by the discovery of the integrability of the Korteweg–de Vries equation. New ideas appeared at that time (such as, for instance, the concepts of Lax and bihamiltonian representations, and the notions of Poisson manifold, Poisson pencil, and recursion operator) which have changed our way of seeing the geometry of integrable systems. They also changed our way of seeing the geometry of separable systems. The paper is an attempt to explain why and in what form the idea of recursion operator leads to reshape the definition of separable system, and to discover a new set of separability conditions called “Kowalevski separability conditions”.

Preface to the special issue: Orthogonal polynomials and integrable systems, Peakons, Toda lattices and Painlevé transcendents

Since the 1990s, the theory of orthogonal polynomials has been increasingly playing an essential role in integrable systems, in particular Toda type lattices, peakon dynamical systems of Camassa–Holm (CH) type, as well as Random Matrix theory, and Painlevé Transcendents. This special issue focuses on recent advances in these areas. Still, in addition to research papers, it also includes review articles providing an overview of the pertinent developments which have impacted the synergy of the theory of integrable systems, approximation theory, and the theory of orthogonal polynomials and generalizations.

Topological Manin pairs and (n,s)-type series

Lie subalgebras of L = {mathfrak {g}}(!(x)!) times {mathfrak {g}}[x]/x^n{mathfrak {g}}[x] , complementary to the diagonal embedding Delta of {mathfrak {g}}[![x]!] and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series {mathfrak {g}}[![x]!] . In this work we consider arbitrary subspaces of L complementary to Delta and associate them with so-called series of type (n, s) . We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s) -type series and topological quasi-Lie bialgebra structures on {mathfrak {g}}[![x]!] . Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems.

Hermite–Padé approximation and integrability

We show that solution to the Hermite–Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of its adjoint linear problem. Our result explains the appearance of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite–Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite–Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.

Bifurcation of Some Novel Wave Solutions for Modified Nonlinear Schrödinger Equation with Time M-Fractional Derivative

In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

Relativistic perfect fluids: integrability of the classical Taub system

It is established that the classical Taub system governing isentropic planar motions of perfect fluids in special relativity may be formulated entirely in soliton-theoretic terms. The well-established tools of integrable systems theory may therefore be used to produce large classes of solutions given in parametric form. As an illustration, the general solution of the Taub system associated with a privileged pressure-density constitutive law is generated by applying a binary Darboux transformation to the seed solution of the Taub system descriptive of extreme fluids.

Unbounded generalization of logarithmic representation of infinitesimal generators

The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution operators, where unboundedness of evolution operator is an essential ingredient of nonlinear analysis. In conclusion, a general framework for the identification between the infinitesimal generators with evolution operators is established. A mathematical framework for such an identification is indispensable to the rigorous treatment of nonlinear transforms, for example, transforms appearing in the theory of integrable systems.

Symmetric matrices and maximal Nijenhuis pencils

A Nijenhuis pencil is a linear subspace of the space of $(1,1)$ tensor field which consists of Nijenhuis operators. The problem of the description of maximal (by inclusion) Nijenhuis pencils containing a subpencil of dimension $n(n+1)/2$ such that the operators in it are - in some system of coordinates - constant symmetric matrices, is solved. Two such pencils turn out to exist, both of which arise in a natural way in applications, for example, in the theory of infinite-dimensional integrable systems. Bibliography: 6 titles.

Циклические фробениусовы алгебры

In this paper, we introduce the notion of cyclic Frobenius algebras (CF-algebras). Canonical structures of CF-algebras exist on associative and Poisson algebras. It turns out that the modern theory of integrable systems yields non-trivial examples of CF-algebras. In the theory of the KdV hierarchy there is a structure of CF-algebra which leads to explicit expressions for the first integrals of the N-th Novikov hierarchy, suitable for classical, non-Abelian and quantum cases.

A behavioural system of strategy‐as‐practice for grant acquisition in nonprofits: An investigation through the sensemaking theory and systems theory

AbstractThe extant literature in strategy‐as‐practice (SAP) has examined it from the sensemaking theory but did not look at it from the system's standpoint. This exploratory study examines the SAP of nonprofits for grant acquisition from a systems perspective by integrating systems theory and sensemaking theory. The grounded theory approach was adopted to gather information and process data, and examine three cases over 3 years. Following the systems approach, the study found sense‐as‐stimuli as the input, sense‐as‐cognitive and behavioural interplay as the process and sense‐as‐strategizing as the output in the sensemaking of strategy‐making in nonprofits for grant acquisition. Sense‐as‐strategizing was also influenced by sense‐as‐cognitive and behavioural interplay in the feedback loop. The strategy‐making system was influenced by (1) the competitive environment for the organization and (2) social environment of the organization. Sense‐as‐cognitive and behavioural interplay influenced the social environment of the organization in a feedback loop.